Question: You have found the following ages (in years) of 5 turtles. The turtles are randomly selected from the 38 turtles at your local zoo: $ 103,\enspace 2,\enspace 1,\enspace 79,\enspace 25$ Based on your sample, what is the average age of the turtles? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 38 turtles, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $5$ samples and divide by $5$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{3721} + {1600} + {1681} + {1369} + {289}} {{5 - 1}} $ $ {s^2} = \dfrac{{8660}}{{4}} = {2165\text{ years}^2} $ We can estimate that the average turtle at the zoo is 42 years old. There is a variance of 2165 years $^2$.